# What is a recursive formula for a geometric sequence?

Oct 28, 2016

Recursive formula for a geometric sequence is ${a}_{n} = {a}_{n - 1} \times r$, where $r$ is the common ratio.

#### Explanation:

A geometric series is of the form

$a , a r , a {r}^{2} , a {r}^{3} , a {r}^{4} , a {r}^{5.} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

in which first term ${a}_{1} = a$ and other terms are obtained by multiplying by $r$.

Observe that each term is $r$ times the previous term. Hence to get ${n}^{t h}$ term we multiply ${\left(n - 1\right)}^{t h}$ term by $r$

i.e. ${a}_{n} = {a}_{n - 1} \times r$

This is called recursive formula for geometric sequence.

There is also explicit formula for ${n}^{t h}$ term i.e. ${a}_{n} = a \times {r}^{n - 1}$